Cart Striking Spring: Velocity & Equations

  • MHB
  • Thread starter Dustinsfl
  • Start date
  • Tags
    Cart Spring
In summary, a cart starts from a height h and goes through a loop of radius r before striking a spring with spring constant k. The equation of motion for the cart strike is m\ddot{x} = -kx, and no other forces need to be considered. The energy of a compressed spring is equal to the initial gravitational energy, and using this equation, we can find the distance the spring is depressed, which is x ≥ √(5mgr/k) if the cart stays in contact with the loop while going through it.
  • #1
Dustinsfl
2,281
5
A cart starts from a height h and goes through a loop of radius r. After going through the loop the cart stikes a spring with spring constant k. How for is the spring depressed.

I have solved for the velocity at the top of loop, \(v^2 = gr\).
I am stuck with coming up with the equation of motion for the cart strike.
\[
m\ddot{x} = -kx +\text{veloctiy?}
\]
do I have \(\sqrt{gr}\) as well? If not, are there any other forces to consider?
 
Mathematics news on Phys.org
  • #2
dwsmith said:
A cart starts from a height h and goes through a loop of radius r. After going through the loop the cart stikes a spring with spring constant k. How for is the spring depressed.

I have solved for the velocity at the top of loop, \(v^2 = gr\).
I am stuck with coming up with the equation of motion for the cart strike.
\[
m\ddot{x} = -kx +\text{veloctiy?}
\]
do I have \(\sqrt{gr}\) as well? If not, are there any other forces to consider?

The equation of motion is:
\[
m\ddot{x} = -kx
\]
And no, there are no other forces to consider once the cart is coming out of the loop.

But rather than using the equation of motion, I would use that the energy of a compressed spring is $\frac 1 2 k x^2$, which must be equal to the initial gravitational energy $mgh$.

Btw, note that it is not given that the cart starts from the top of the loop. It is only given that it starts at height $h$.
 
  • #3
I like Serena said:
The equation of motion is:
\[
m\ddot{x} = -kx
\]
And no, there are no other forces to consider.

But rather than using the equation of motion, I would use that the energy of a compressed spring is $\frac 1 2 k x^2$, which must be equal to the initial gravitational energy $mgh$.

Btw, note that it is not given that the cart starts from the top of the loop. It is only given that it starts at height $h$.

The height h is a point before the loop. \(h\geq \frac{5}{2}r\) in order to make it through the loop.

So you are saying use the conversation of energy equation.
\[
mgh_i+ 0\cdot KE = mgh_f + \frac{1}{2}kx^2
\]
What would \(mgh_f\) be though?
 
  • #4
dwsmith said:
The height h is a point before the loop. \(h\geq \frac{5}{2}r\) in order to make it through the loop.

So you are saying use the conversation of energy equation.
\[
mgh_i+ 0\cdot KE = mgh_f + \frac{1}{2}kx^2
\]
What would \(mgh_f\) be though?

I assume $h$ is the height above the spring.
 
  • #5
I like Serena said:
I assume $h$ is the height above the spring.

Yes. The spring would be h = 0 after the cart goes through the loop.

So is it
\[
mgh = \frac{1}{2}kx^2\Rightarrow x = \sqrt{\frac{2mgh}{k}}
\]
but using the fact that \(h\geq \frac{5}{2}r\), we have
\[
x\geq\sqrt{\frac{5mgr}{k}},
\]
correct?
 
Last edited:
  • #6
dwsmith said:
Yes. The spring would be h = 0 after the cart goes through the loop.

So is it
\[
mgh = \frac{1}{2}kx^2\Rightarrow x = \sqrt{\frac{2mgh}{k}}
\]
but using the fact that \(h\geq \frac{5}{2}r\), we have
\[
x\geq\sqrt{\frac{5mgr}{k}},
\]
correct?

Yes.
That is assuming that the cart stays in contact with the loop while going through it.
It seems that that piece of information is not given.
 

FAQ: Cart Striking Spring: Velocity & Equations

What is the Cart Striking Spring experiment?

The Cart Striking Spring experiment is a physics experiment used to study the relationship between velocity and spring constant. It involves a cart being launched towards a spring and measuring the resulting compression of the spring.

What equipment is needed for the Cart Striking Spring experiment?

The equipment needed for this experiment includes a cart, a spring, a track for the cart to roll on, a timer, and a measuring tool to record the displacement of the spring.

How does the velocity of the cart affect the spring's compression?

The greater the velocity of the cart, the greater the compression of the spring will be. This is because the kinetic energy of the cart is transferred into potential energy stored in the spring, causing it to compress.

What is the equation for calculating the spring constant in this experiment?

The equation for calculating the spring constant is k = (m*g*d)/x, where k is the spring constant, m is the mass of the cart, g is the acceleration due to gravity, d is the displacement of the spring, and x is the initial compression of the spring.

What are some sources of error in the Cart Striking Spring experiment?

Some sources of error in this experiment include friction in the cart's wheels, variations in the spring constant due to temperature changes, and human error in measuring the displacement of the spring.

Back
Top